A226901 Partial sums of Hooley's Delta function.

1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 19, 20, 22, 24, 26, 27, 29, 30, 33, 35, 37, 38, 42, 43, 45, 46, 48, 49, 52, 53, 55, 56, 58, 60, 63, 64, 66, 67, 71, 72, 75, 76, 78, 80, 82, 83, 87, 88, 90, 91, 93, 94, 96, 98, 101, 102, 104, 105, 109, 110, 112, 114, 116, 118, 120, 121

Offset: 1

Author

Charles R Greathouse IV, Jun 21 2013

Keywords

• nonn

Comments

Tenenbaum (1985) proves that a(n) < n exp(c sqrt(log log n log log log n)) for some constant c > 0 and all n > 16. Numerically, c appears to be close to 0.5 or 0.55.

References

• R. R. Hall and G. Tenenbaum, On the average and normal orders of Hooley's ∆-function, J. London Math. Soc. (2), Vol. 25, No. 3 (1982), pp. 392-406.
• C. Hooley, On a new technique and its applications to the theory of numbers, Proc. London Math. Soc. 3 38:1 (1979), pp. 115-151.
• Gérald Tenenbaum, Sur la concentration moyenne des diviseurs, Commentarii Mathematici Helvetici 60:1 (1985), pp. 411-428.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Dimitris Koukoulopoulos and Terence Tao, A note on the mean value of the Erdős-Hooley Delta function, arXiv preprint (2023). arXiv:2306.08615 [math.NT]

Crossrefs

Partial sums of A226898.

Cf. A226899, A226900.

Programs

Maple

with(numtheory):
b:= n-> (l-> max(seq(nops(select(x-> is(x<=exp(1)*l[i]), l))-i+1,
        i=1..nops(l))))(sort([divisors(n)[]])):
a:= proc(n) a(n):= b(n) +`if`(n=1, 0, a(n-1)) end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 21 2013

Mathematica

delta[n_] := Module[{d = Divisors[n], m = 1}, For[i = 1, i < Length[d], i++, t = E*d[[i]]; m = Max[Sum[Boole[d[[j]] < t], {j, i, Length[d]}], m]]; m];
A226901 = Array[delta, 100] // Accumulate (* Jean-François Alcover, Mar 24 2017, translated from PARI *)

PARI

Delta(n)=my(d=divisors(n), m=1); for(i=1, #d-1, my(t=exp(1)*d[i]); m=max(sum(j=i, #d, d[j]
				

Formula

n log log n << a(n) << n (log log n)^(11/4); the lower bound is due to Hall & Tenenbaum (1988) and the upper bound to Koukoulopoulos & Tao.